T1 space

A topological spaceMathworldPlanetmath (X,τ) is said to be T1 (or said to hold the T1 axiom) if for all distinct points x,yX (xy), there exists an open set Uτ such that xU and yU.

A space being T1 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the following statements:

  • For every xX, the set {x} is closed.

  • Every subset of X is equal to the intersectionMathworldPlanetmath of all the open sets that contain it.

  • Distinct points are separated.

Title T1 space
Canonical name T1Space
Date of creation 2013-03-22 12:18:14
Last modified on 2013-03-22 12:18:14
Owner drini (3)
Last modified by drini (3)
Numerical id 10
Author drini (3)
Entry type Definition
Classification msc 54D10
Synonym T1
Related topic T0Space
Related topic T2Space
Related topic T3Space
Related topic RegularSpace
Related topic ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA
Related topic SierpinskiSpace
Related topic PropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2